具有回火α-Stable等待时间的分数阶Fokker-Planck方程

The Fractional Fokker-Planck Equation with Tempered α-Stable Waiting Times

摘要: 本文介绍了一种具有回火稳定等待时间(tempered stable waiting times)的朗之万型次扩散(Langevin-type subdiffusion)模型。我们考虑了空间相关外力场的情况。该模型在小时间尺度上表现出次扩散行为，并在大时间尺度上收敛到正常扩散。我们从回火稳定过程(tempered stable processes)的理论推导出回火反常扩散(tempered anomalous diffusion)的一般性质，特别是推导了对应于回火次扩散的分数阶福克–普朗克方程(fractional Fokker-Planck equation)的形式。此外，我们构建了一种算法来模拟所引入过程的样本路径。我们应用该算法来近似求解分数阶福克–普朗克方程，并使用蒙特卡洛方法(Monte Carlo methods)研究回火次扩散的统计特性。

Abstract: In this paper, we introduce a Langevin-type subdiffusion model with tempered stable waiting times. We consider the case of a spatially dependent external force field. The model exhibits subdiffusion behavior at small time scales and converges to normal diffusion at large time scales. We derive the general properties of tempered anomalous diffusion from the theory of tempered stable processes, particularly identifying the form of the fractional Fokker-Planck equation corresponding to tempered subdiffusion. Additionally, we construct an algorithm to simulate sample paths of the introduced process. We apply this algorithm to approximate solutions of the fractional Fokker-Planck equation and investigate the statistical properties of tempered subdiffusion using Monte Carlo methods.